First, some terminology. 15mm and up is still just "wide angle". It's below 15mm that lenses become ultrawide. (More at What is an ultra-wide lens?) So if you want ultrawide, you need lenses that are around the 10-15mm focal length range. Wide angle on a 1.6x crop camera, typically means something in the 15-24mm range. Normal is around 28-35. And telephoto begins around 40mm.
Whether you need a wide or an ultrawide is a matter of personal taste, but essentially going very wide means more distortion as well as more coverage, so you're basically choosing how funky and extreme you want to get with the wide angle. Also, be aware that most of these wide angle choices on Canon are only for crop bodies, and probably will have to be replaced if you ever plan to make the full-frame move.
Whether you need stabilization or a large max. aperture depends on how you plan to use the lens. Most folks do not find IS to be critical with ultrawide lenses, because their main use will be for landscape/cityscape shooting with a tripod. And a tripod will be IS any day when it comes to long exposures. While it is more inconvenient, it also is a much better tool than IS for long exposures, and generally doesn't have an upper limit, shutter-speed-wise, like IS does.
In addition, good handholding technique and choosing appropriate exposure settings can take care of camera shake issues as much as IS, and IS, while convenient, is still no guarantee of eliminating motion blur from camera shake.
The only use where you might want stabilization and a large max. aperture is if you plan to shoot environmental portraiture handheld, or architectural interiors without lighting.
That said, there is one lens that it seems you have not considered which is both 10mm and has image stabilization, while being lower-cost than either of the two lenses you're looking at: the EF-S 10-18 f/3-5.6 IS STM.
Also, if you find that you are an ultrawide junkie and you'd like to go even wider, there is one class of lenses you might want to consider if the distortion doesn't scare you off, and that would be fisheye lenses. Fisheyes give up attempting to project the image in rectilinear fashion (i.e., keeping straight lines straight), and allows lines to curve (maps equisolid or stereographic). I shot for many years without an ultrawide because I had a fisheye in the bag.
